Degenerate two-variable $ q $-Legendre polynomials via $ q $-operational calculus and degenerate Laplace/Sumudu transforms

We introduce a degenerate $ (q, \lambda) $-extension of the two-variable $ q $-Legendre-type polynomials by deforming the $ q $-Bessel–Tricomi kernel through the degenerate falling-factorial weights $ (1)_{k, \lambda} $. The resulting family $ \{\mathcal{L}^{(\lambda)}_{n, q}(x, y)\}_{n\ge 0} $ is defined by a single generating function and interpolates both the recently studied two-variable $ q $-Legendre polynomials and their classical limits as $ \lambda\to 0 $ and/or $ q\to 1 $. We derive an explicit finite-sum representation, an operational Rodrigues-type formula, and a quasi-monomial structure that yields raising and lowering operators together with a fundamental $ q $-difference equation in the variable $ y $. We further compute the degenerate Laplace and Sumudu transforms of the generating kernel and obtain corresponding transform identities for $ \mathcal{L}^{(\lambda)}_{n, q} $. Several reduction formulas, even/odd subsequence decompositions, and a moment-functional interpretation are presented, along with linearization coefficients, low-degree examples, and numerical table.